Characterizes when a family of continuous functions on a compact space has a uniformly convergent subsequence, in terms of the family being uniformly bounded and equicontinuous. It is a central compactness criterion in functional analysis, named for Cesare Arzela and Giulio Ascoli.
Facts
StatementIf a sequence of real-valued continuous functions on a closed and bounded interval is uniformly bounded and uniformly equicontinuous, then it has a subsequence that converges uniformly. 1 Proof Method Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Principal Result Of
Source Cesare Arzelà (Wikipedia)
Source Giulio Ascoli (Wikipedia)
Proved By
Source ProofWiki: Arzelà-Ascoli Theorem
Source ProofWiki: Arzelà-Ascoli Theorem
Sources
1. Arzela-Ascoli Theorem (Wikipedia)
Wikimedia FoundationStatement and first consequences
If this sequence is uniformly bounded and uniformly equicontinuous, then there exists a subsequence {fnk}k ∈ N that converges uniformly.
lead section, history sentence
A weak form of the theorem was proven by Ascoli (1883-1884), who established the sufficient condition for compactness, and by Arzelà (1895), who established the necessary condition and gave the first clear presentation of the result.
View the Source 2. ProofWiki: Arzelà-Ascoli Theorem
ProofWikiProof section
Aiming for a contradiction, suppose $F$ is relatively compact, but not pointwise equicontinuous.
Proved By: Cesare Arzelà, Lead sentence
The notion of equicontinuity was introduced in the late 19th century by the Italian mathematicians Cesare Arzelà and Giulio Ascoli.
Proved By: Giulio Ascoli, Lead sentence
The notion of equicontinuity was introduced in the late 19th century by the Italian mathematicians Cesare Arzelà and Giulio Ascoli.
View the Source Cesare Arzelà (Wikipedia)
Principal Result Of: Cesare Arzelà, Lead, Arzela-Ascoli theoremQuote, Principal Result Of: Cesare Arzelà, Lead, Arzela-Ascoli theorem
in the Arzelà-Ascoli theorem
Giulio Ascoli (Wikipedia)
Principal Result Of: Giulio Ascoli, Body, contributionsQuote, Principal Result Of: Giulio Ascoli, Body, contributions
In 1889, Italian mathematician Cesare Arzelà generalized Ascoli's Theorem into the Arzelà-Ascoli theorem, a practical sequential compactness criterion of functions.
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